Arithmetic

I am teaching arithmetic this term.

It is not ordinary grade school arithmetic. Instead, my students are revisiting arithmetic with fresh eyes.

Here’s what we will do:

• Learn arithmetic again. Kind of regular arithmetic – except instead of 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11,… we will use /, ∆, ☐, /❍, //, /∆, /☐, ∆❍, ∆/, ∆∆, ∆☐,  ☐❍,… I will call this “the abnormal number system”
• Sneak in some new concepts and definitions. For example, we will dwell at length on the idea of “what comes next” and give it a name – each number has a “successor.” We will attempt to create formal definitions for addition and multiplication, and blend them with our work.
• Teach a basic outline of the 19th Century history of the axiomatization of arithmetic

Here the seminars will diverge. There are two seminars – one for 9th and 10th graders. The other is for 11th and 12th grades. It’s the latter that will forge ahead further, while the former might just get a taste.

• Review proof by contradiction (all of the juniors and seniors previously had a seminar in Set Theory or Logic with some substantial indirect proof. This is review).
• Learn some simple proof by induction. Do some standard high school level problems. Though I am not sure if ‘standard”\’ is the right word – it seems that very few high school students see any proof by induction at all.
• Construct the Natural Numbers, using a modified version of how it was done in the late 1800s. This is Very Hard. We will use the first chapter (10 pages) of this text, reading every line, doing every exercise, completing every proof*.

How was it done in the 1800s?

Warning – gross oversimplification coming: In the 19th Century some mathematicians wondered why Geometry has Postulates, and Undefined Terms, and Formal Definitions, and Theorems, and Corollaries, and Lemmas – but Arithmetic does not. And so they did the obvious – they created Postulates, and Undefined Terms, and Formal Definitions, and Theorems, and Corollaries, and Lemmas – all for Arithmetic. The postulates are called Peano Postulates, after Giuseppi Peano.

A couple more things:

The two seminars have fourteen kids each (just worked out that way). They meet once a week each (by design). I am requiring reading and some exercise outside of class, for preparation. I am not grading homework, and there are no tests or quizzes. We are meeting to discuss mathematics, to ask questions, to explore new terms and notation, new ways of looking at things that once seemed familiar, and to consider what it means to prove something.

An instructor defined proof as “that which convinces.” Maybe I heard that in 1996 or 1998. I am convinced (ha ha) that I have not encountered a better definition. Ever. I share it with my own students.

The instructor was David Rothchild. I say “instructor” because that is what he was. David was quick to correct anyone who called him “professor” or “doctor” – he did not have a PhD. I met him at Lehman College, where I finished my undergraduate studies, and began my graduate work.

In the mid and late 90s I sat with Rothchild for four math courses, courses for aspiring teachers of mathematics. In 601, I think it was, he taught us to view arithmetic with fresh eyes, through a system he called “abnormal arithmetic.” He used four symbols: a slash, a triangle, a square, and a circle. He snuck in concepts, such as “successor” that we would use later in the course – during those harder parts when he introduced Peano’s Postulates. Few courses have had a greater impact on my decades of teaching mathematics.

Two years ago I channelled Rothchild, and tried my own “Axiomatic Arithmetic” – based on my memory of his course from over 20 years earlier. It worked well. That seminar started in person in February of 2020. We suspended it for a few weeks when the pandemic hit, but moved to remote, and met every week for an hour of pure math, until late June. An alum, now a math major at a prestigious institution wrote “I was behind much of the class when it comes to content but I’m pretty far ahead when it comes to proofs. Axiomatic arithmetic was honestly the best way I possibly could have prepared…”

This year I am trying to do it a little better. We will be in person for the full seminar. That’s a plus. And there has been some hype about how challenging it might be, and what an accomplishment it will be to complete this. Eyes wide open. Very eager eyes. It helps.

I will write up synopses of some of the seminars. I will get to ‘successors’ later this week. Maybe ‘predecessors,’ too. The older kids have added. The younger will get their chance after break.

I’ll show you, as I we go. This is fun.